1. Let's start by stating the problem: We want to understand the concept of "distance" in geometry, which is the measure of how far apart two points are. 2. The formula to find the distance between two points $A(x_1, y_1)$ and $B(x_2, y_2)$ in a plane is derived from the Pythagorean theorem: $$\text{Distance} = d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. Important rules: - The distance is always positive or zero. - Squaring the differences ensures all values are positive. - The square root gives the actual length. 4. Let's explain with an example: Suppose $A(1, 2)$ and $B(4, 6)$. Calculate the differences: $$x_2 - x_1 = 4 - 1 = 3$$ $$y_2 - y_1 = 6 - 2 = 4$$ Square these differences: $$3^2 = 9$$ $$4^2 = 16$$ Add them: $$9 + 16 = 25$$ Take the square root: $$\sqrt{25} = 5$$ So, the distance between points $A$ and $B$ is 5 units. 5. In simple terms, distance measures how far two points are by forming a right triangle between them and using the Pythagorean theorem to find the length of the hypotenuse. This is the basic concept of distance in grade 9 geometry.